Part I: The biology and the replication globes
I.1.1 “It has become almost a cliché to remark that nobody boasts of ignorance of literature, but it is socially acceptable to boast ignorance of science and proudly claim incompetence in mathematics”. Richard Dawkins.
This paper uses the DNA genetic code with its nucleotide codon syntax and semantics in combination with a four-dimensional hypersphere and the well-known invertible topological properties of a Möbius strip to thoroughly refute contemporary biology’s general consensus that universal laws and deductive logic are inapplicable, if not impossible, because biological phenomena are “too complex”. We instead show that topology’s “classification theorem” allows us to place all biological structures, entities, and populations into four equivalence classes based on the sphere, the torus, and the real projective plane. We thereby provide the four laws of biology, the four maxims of ecology, and the three constraints to which all biological entities and populations are subject.
I.1.2 Our model begins by perusing the DNA genetic code … but is not restricted to that bioinformatics approach. As in Figure 1, it reviews all biological-ecological events, assigning them to at least one of two distinct globes of bioactivity.
We provide one “biology globe”. This causes all the more generally biological behaviours. They are the “recurvature” interactions for a generation. We also provide a more specifically “replication globe”. It is responsible for all more explicitly reproductive recurvatures.
Each of our two globes or spheres of activity contributes to an overall biological generation by recurving events about its surface and around its interior and central point. Since each such globe has the potential to create a “circulation” about itself, then these various elements, taken together, allow us to declare that biological populations are “infinite cyclic groups”, with infinite capabilities. They come complete with their equally infinite cyclic subgroups. They can each generate infinitely many such distinct groupings and subgroupings, each over its potentially infinitely many generations. These are the manifold species.
I.1.3 Our four dimensions arise because our biology and our replication globes allow for four movements. Three are similar to the ordinary three dimensions of space. They cohere around the central point in each. The fourth is their movement towards and away from each other.
Just as a developing hurricane builds a “wind wall” all about itself, so also are all biological macromolecules, entities, and populations similar kinds of wind walls. They result from the recurvature movements about our two biology and replication globes. The recurvatures are then expressed in terms of their locations about each globe as latitudes; as longitudes; and as heights above each; as well as in terms of their overall circulation lengths and rates as regards those two globes moving towards and away from each other.
I.1.4 Our model imports the Chomsky(–Schützenberger) hierarchy’s power to conceptualize four main groups of biological actions. The infinite cyclic groups, subgroups, and wind wall recurvatures create the four main categories of biological artefacts:
• some are biological but fail to replicate;
• others are both biological and replicative;
• yet others not only replicate, but add a host of ancillary but non-replicative biological activities;
• a final group—including Homo sapiens—are entirely replicative, but give the impression of also being biological.
Our model shows that the four groupings listed above are inevitable results of DNA’s bioinformatic and genetic syntaxes and semantics. The biological Chomsky grammar our model establishes states the recurvature rules of selected sets of chemical components—moving as wind walls—about the two globes.
I.1.5 Our Chomsky grammar and its resulting hierarchy provide the backbone for the four-dimensional and topological framework in Figure 1. The four rows assign biological recurvatures to the two globes in accordance with their codings. Those produce the accelerative and decelerative behaviours that create the infinite cyclic groups and subgroups in terms of their distances about the globes for the four types.
I.1.6 Our model’s Chomsky grammar reflects the underlying DNA genetic language. A part of its power is the realization that it is always possible to find some cylinder of maximum volume, V, that can replace any sphere. The cylinder’s more rectilinear surfaces, S, give it the potential to turn and to extend infinitely outwards at each point. It can then roll as far in any direction as any sphere.
But if a generation is to complete, then there must always be recurvatures about the globes. Every biological interaction that emerges through some surface, S, originates in some volume, V, that also guarantees those continuing recurvatures. We can therefore describe a grammar for those V behaviours through those S presenting surfaces.
I.1.7 Our model thoroughly exploits the realization that a Chomsky grammar is both abstract and versatile. The latter’s dominion can be applied to any field desired. It can for example explain an urban sprawl. The relationship between our globes, our surfaces, and our volumes is then similar to that between a language, a grammar, its syntax, and its semantics.
Given that a Chomsky-style grammar describes an entire living language, L, it is the surprisingly concise:
[Σ, S, δ, α0, F],
where Σ is a finite input alphabet of discrete symbols that can represent any object desired; S is the set of possible alphabet combinations; δ is some “transition function” or set of “production rules” over the Cartesian product Σ × S; α0 is an initial state; and F is the set of permissible terminators in that language.
By convention, the intermediary nonterminals, constructors, and indicators that build a language are represented by upper case letters such as S and F; its completed sentences—or more strictly, its “terminals”—are represented by lower case letters; while Greek letters represent strings of both. So if it is our present desire to compile an English dictionary, then our input alphabet will (at its simplest) be the standard English one of ‘a’, ‘b’, ‘c’. A set of production rules similar to A → a will then produce all possible English words for its terminals. But we can use the same structure and step up a hierarchy to let our input alphabet instead be the entire English lexicon. Our nonterminals will then be phrase structures. These can combine through a set of production or grammatical rules to create all possible well-formed English sentences. And with appropriate structures and rules, we can describe a town.
I.1.8 Biological processes and language are both highly complex. No given—and necessarily simple—example can convey the sophistication possible to a Chomsky grammar. But we can nevertheless consider a language, L, in which:
• our finite alphabet has only two symbols, and so that our input alphabet Σ = {a, b};
• our sole initiating nonterminal, α0, is A;
• we have a Rule 1 production or transformation of A → ba;
• and a Rule 2 of A → aAb.
We can now take up some A as an α0 to begin a production. We can next pick either of the two rules. If we pick Rule 1, we get the trivial production A → ba. Since this is all terminals, we are done.
We can begin again. This time, our initiating α0 production is Rule 2. We now have A → aAb. Since this is a mixture of terminals and nonterminals, we invoke Rule 2 again, to substitute for that A. We get aAb → aaAbb. Invoking Rule 2 yet again gives aaaAbbb. And if we now pick Rule 1, we terminate with aaababbb.
This rudimentary grammar over L, with its only two rules and its two alphabet symbols, allows us to create all strings, of arbitrary length, of the general form αbaβ where α and β are equal-length strings of as and bs respectively. Its significance and meaning depend upon the field over which this is a language. If, for example, it is an architecture, then we are describing the different kinds of neighbourhoods in an urban sprawl.
I.1.9 Chomsky productions and grammatical rules are considerably more sophisticated but—similarly to strings of codons—they all produce strings of terminals and nonterminals. Their rules therefore fall into the four broad and well-known types of 3, 2, 1, and 0 that are the four grammars in the Chomsky hierarchy. Our model then assigns DNA’s biological–ecological productions to one or another of the four rows in Figure 1. They represent the transformation rules that distinguish between the different fertility spaces, fertility groupings, and fertility recurvatures that DNA uses to replicate; and also between the biological spaces, groupings, and recurvatures that the replicated entities then use to interact with the surroundings.
I.1.10 In the language of information science, DNA’s Chomsky style productions are physically and quantitatively measurable upon our globes. Both sets of the biological and the fertility materials are countable. They are each equipollent with ℵ0, the set of countably infinite natural numbers (Weisstein 2015a).
I.1.11 The four groupings on the left in Figure 1 are the set of non-replication-based recurvatures about our biology globe. They are oriented “trivial cycles”. They bound definite regions on the globe’s surface, so creating the wind walls that surround their centres of activity.
The four groupings on the right differ by being reversible journeys about a Möbius strip. They do not create wind walls. They instead impose curvatures and accelerations. They create the recurvatures about our replication globe.
I.1.12 The genetic code requires both a transmitter and a receiver. There must be both a wind wall and a velocity upon one or both globes.
I.1.13 The successful transmission of hereditary information requires both (a) a syntax, and (b) a semantics. This is the propagation of discrete materials, and their circulating wind wall, about both of our globes. The receiver must therefore possess (a) a cipher to decode the necessary information, and (b) the structures that permit it to carry out whatever task is communicated … which is in this case to produce viable biological entities by replicating the winds and wind wall that begot it. They must therefore travel successfully about our two globes. We refer to the Chomsky production rules that achieve this recurvature, and on the right of Figure 1, as a “recursive function”, δ.
I.1.14 As again in Figure 1, the recursive functions in our model have their complementarities of “reversions” and “derivations”. They build our recurvatures about the globes, and from their initial symbols to their terminal ones for a generation. Every replication phenomenon derives from a recursive function journey about a Möbius strip. It is some hyperspherical “biovolume”, V, that can independently travel about our globes.
But that journey is not just a recurvature about the globes. It is also some associated biological presentation. That surface presentation is the “biosurface”, S. Its enclosed biovolumes as codons use their production rules to construct biological effects. The resulting biosurfaces, also codons, are their observed biological presentations. Since those surface presentations are, necessarily, terminals, we refer to them as “loops”.
I.1.15 Our model builds on the fact that DNA, acting as a genetic code, stores information in its codons of three nucleotides each. Genes are the functional segments, having four possible bases. The three nucleotides therefore give 43 = 64 different possibilities. Their combinations specify the 20 different amino acids used by all living organisms.
I.1.16 The use of a formal code, of this type, to accomplish such a purpose again requires the receiver of the code to understand the syntax and its rules, and to accord the correct meaning in order to accomplish either, or both, of the biological and/or fertility tasks implied by those symbols. The biovolumes, V, that DNA generates thus contain biological energies per some unit volume. As recursive functions, they are productions that are then incident upon their presenting biosurfaces as terminals. Their successful transfer thus begets some momentum per relevant unit area, expressed as a timed loop.
I.1.17 It is unfortunately impossible to represent the four dimensions DNA uses to construct its biological hypersphere in the only two available to us on a piece of paper. Nevertheless, and as in Figure 2, our model will represent a biological generation as two globes. One can initially be regarded as nesting inside the other as they roll. The outer biology globe has the behaviours that supervise all biological interactions and recurvatures; while the inner replication one exclusively handles replications, originating all recursive functions.
The outer biology globe interacts directly with the surroundings to construct our DNA nucleotide codons using three dimensions equivalent to the left-right, front-back, and up-down of ordinary physical space. We can initially think of the fourth dimension—most generally known as ‘upsilon’–‘delta’—as a combination of a horizontal rolling over some absolute time period, T, and an internal pulsing, during that rolling, to and from each other of the inner replication and outer biology globes. They do so over some distance, τ. That combination of τ–T is a joint transition (a) to and from the replication point; and (b) forwards in time from the beginning to the end of a generation.
Our model gets its power by using topology’s classification theorem for compact surfaces to examine the conjoined surface these two globes create. All biological structures then fall into very specific equivalence classes based on the resulting normal forms they establish.
Our horizontal hyperglobe translation of time T, combined with our waxing and waning of distance τ, produce the complete set of DNA and biological-ecological interactions, λ, for a recurvature and a generation. Our hyperglobe achieves this by drawing in the needed resources to build its Möbius strip based Vn molecular reversions. Some of those are allocated to the biology globe, others to the replication one, and yet others to both.
All such “lifts” up into our linked globes, from the surroundings, are uniquely determined by their initiating values at any point. The productions are therefore some hyperspherical Vn. This is only sometimes replicative; but is always intrinsically biological. The constructed materials now extrude and loop rectilinearly outwards, from the globes, as derivations. They thus produce a set of observed Sn-1 biosurfaces and events. Biology therefore consists of this Sn-1–Vn or else Sn–Vn+1 coupling, with a replication point, ρ, acting as a pole.
I.1.18 The cells at bottom right in Figure 1 are V1 Möbius strip style biovolumes relative to the S0 molecules on their left that “tangle” to create their base pairs … with those same molecules manifesting the relevant biological behaviours in the surroundings as their presenting biosurfaces. But the manifest molecules have a parallel upwards recursion and complexity. They seem able to come together to build the observable S1 genomes above them. But those apparent productions are merely the biosurfaces belonging to the parallel reversion, on the right, of cells into the V2 biovolumes that are the enshrining entities for those same genomes. Those are built by our recursive functions.
The DNA codon grammar in our parallel structure is that only those biological constructs that pierce the inner replication globe and return, to interact with the surroundings, have the appropriate syntax and semantics. Only they are successfully fertile and replicative.
I.1.19 Our grammar stipulates that the period between the beginning of a generation and the pole or replication point is a “fibration”, θ; with that between the replication point and the end of the generation being the “cofibration”, ρ. The fibration takes in resources and energy from the surroundings; the cofibration then returns them to complete the generation.
I.1.20 As in Figure 3, the language that DNA speaks is that reproduction is a “mapping” of the form φ:X → Y. DNA maps from a first group of elements in a “progenitor domain”, X. Those transform to a second set of elements in a “progeny codomain”, Y. All such φ:X → Y and/or φ:Y → X mappings are our recursive Chomsky production rules, δ.
Our mappings will generally occur via a “mapping cylinder”, Mλ. That mapping cylinder must then impose sufficient accelerations and velocities on both (a) the groups; and (b) the surroundings to create the needed recurvatures.
I.1.21 We also define the progenitor domain, X, as the “preimage”; and the progeny codomain, Y, as the “image”. A “homomorphism”—from the Greek homos for ‘same’, and morphe̅ for ‘form’—is then a structural mapping between the groups contained in these different globes and domains.
A biological homomorphism preserves a group’s essential characteristics. It does so throughout all the group’s possible translations and transformations, and quite irrespective of the biological and/or replicative spaces that group transitions through.
I.1.22 Our model very carefully separates structures from spaces. Since we have our two globes, then we can carefully analyse the realization that not all entities in any one group are obliged to recurve about our two globes at the same velocities and accelerations as they transition across a generation, and so from preimage to image and/or conversely. Entities can therefore map between domain and/or codomain without necessarily also keeping the spaces they each traverse—and so their recurvature values—invariant. They do not all recurve about our two centres in the same ways.
A “homeomorphism”, from the Greek homoios for ‘similar’, is a successful mapping between spaces, rather than between groups of elements. A biological homeomorphism therefore preserves the biological space’s characteristics, irrespective of the groups and/or transformations passing through it.
We shall only regard groups of entities as identical if they preserve both (a) their homomorphism, and so group structure; and (b) their homeomorphism, and so space-behaving characteristics. Any populations that successfully match a generation’s worth of biological activities with a generation’s worth of replicative ones have clearly preserved an appropriate structure across all spaces, and will then be both homomorphic and homeomorphic. This is “isomorphic” where iso comes from the Greek for “identical”.
I.1.23 The significance of so distinguishing between iso-, homo-, and homeomorphism is that, as in Wernicke’s aphasia, DNA syntax transmission over a generation is possible without semantic comprehension by the receiver; while as in Broca’s aphasia, it is possible to transmit the semantic comprehension capability that would make a generation possible, but then to fail to effect it by transmitting no—or the wrong—syntax for the decoding.
I.1.24 DNA’s interactive, complementary, and parallel language and grammar of homo- and homeomorphisms between (a) preimage and image, and (b) biology and replication globes, produces Figure 1’s four horizontal groupings. They are a relationship between our globes. Groups of entities use fibration and cofibration to transition back and forth between progenitor domain and progeny codomain, and so between syntax and semantics.
I.1.25 Each of our four topological style groupings—based as they are on the sphere, the torus, and the real projective plane—is characterized by its “pollency” or “fertility power”, taken from the Latin polle̅re ‘to be strong’. This reflects the relationship between the syntax and semantics that produce the different biology and fertility behaviours across our two globes. Each possible combination—being homomorphic and/or homeomorphic—is now “injective”, , and/or “surjective”, , i.e. “one-to-one” and/or “onto”. The four possible groupings of biological artefacts that our model establishes across all biological Kingdoms are:
I.1.26 We shall now prove that these various and parallel trivial cycle and Möbius strip interactions are DNA’s entire language and grammar of syntax and semantics. They invoke our two globes and our two domains to establish (a) the four maxims of ecology that in their turn establish the homeomorphic spaces that can support the wind walls that recurve about our two globes; (b) the four laws of biology that establish the homomorphic structures that can accelerate and decelerate between the biosurfaces and the biovolumes; and (c) the three constraints that link the homomorphic to the homeomorphic to create the circulations of the generations. We shall take Brassica rapa as a test case.